Relative grading in Heegaard Floer homology ($\mathrm{gr}(\mathbf{x})-\mathrm{gr}(\mathbf{y})$)

For generators in the same Spin-c structure and a Whitney disk class $\psi$ joining them,The ambiguity is the divisibility of $c_1(\mathfrak{s})$ on $H_2(Y)$; a torsion determinant class gives an integer relative grading. Adding a sphere class changes $\mu$ by two and $n_z$ by one, so it leaves the difference unchanged. For a domain $D$, the index can be calculated as $\mu (D)=e(D)+n_{\mathbf{x}}(D)+n_{\mathbf{y}}(D)$, where $e$ is Euler measure and each $n_{\mathbf{x}}$ is the sum of averages of the four adjacent region multiplicities at the selected intersections.

For a torsion Spin-c structure, the relative grading of Heegaard Floer homology lifts canonically to a rational absolute grading, normalized by the bottom plus generator of $S^3$ having degree zero. Multiplication by $U$ has degree $-2$, and a Heegaard Floer cobordism map has degree $(c_1^2-2\chi -3\sigma )/4$. The correction term $d(Y,\mathfrak{s})$ is the lowest grading of the image of $H{F}^{\infty }$ in $H{F}^{+}$ for a rational homology sphere. A pure tower then has bottom plus degree $d$ and top minus degree $d-2$ in the $i<0$ minus convention. Keeping that shift explicit prevents errors of two in surgery and mixed-invariant calculations.